quantum mechanics - Is there something behind non-commuting observables?

Consider a quantum system described by the Hilbert space $\mathcal{H}$ and consider $A,B\in \mathcal{L}(\mathcal{H},\mathcal{H})$ to be observables. If those observables do not commute there's no simultaneous basis of eigenvectors of each of them. In that case, in general if $|\varphi\rangle$ is eigenvector of $A$ it will not be of $B$.

This leads to the problem of not having a definite value of some quantity in some states.

Now, this is just a mathematical model. It works because it agrees with observations. But it makes me wonder about something. Concerning the Physical quantities associated to $A$ and $B$ (if an example helps consider $A$ to be the position and $B$ the momentum) what is really behind non-commutativity?

Do we have any idea whatsoever about why two observables do not commute? Is there any idea about any underlying reason for that?

Again I know one might say "we don't care about that because the theory agrees with the observation", but I can't really believe there's no underlying reason for some physical quantities be compatible while others are not.

I believe this comes down to the fact that a measurement of a quantity affects the system in some way that interferes with the other quantity, but I don't know how to elaborate on this.

EDIT: I think it's useful to emphasize that I'm not saying that "I can't accept that there exist observables which don't commute". This would enter the rather lengthy discussion about whether nature is deterministic or not, which is not what I'm trying to get here.

My point is: suppose $A_1,A_2,B_1,B_2$ are observables and suppose that $A_1$ and $B_1$ commute while $A_2$ and $B_2$ do not commute. My whole question is: do we know today why the physical quantities $A_1$ and $B_1$ are compatible (can be simultaneously known) and why the quantities $A_2$ and $B_2$ are not?

In other words: accepting that there are incompatible observables, and given a pair of incompatible observables do we know currently, or at least have a guess about why those physical quantities are incompatible?